Effect of Variability in Payoffs on Conditions for the Evolution of Cooperation in a Small Population

Abstract

In this paper, we study the effect of stochastic fluctuations in payoffs for two strategies, cooperation and defection, used in random pairwise interactions in a population of fixed finite size with an update according to a Moran model. We assume that the means, variances and covariances of the payoffs are of the same small order while all higher-order moments are negligible. We show that more variability in the payoffs to defection and less variability in the payoffs to cooperation contribute to the evolutionary success of cooperation over defection as measured by fixation probabilities under weak selection. This conclusion is drawn by comparing the probabilities of ultimate fixation of cooperation and defection as single mutants to each other and to what they would be under neutrality. These comparisons are examined in detail with respect to the population size and the second moments of the payoffs in five cases of additive Prisoner’s Dilemmas. The analysis is extended to a Prisoner’s Dilemma repeated a random number of times with Tit-for-Tat starting with cooperation and Always-Defect as strategies. Moreover, simulations with an update according to a Wright–Fisher model suggest that the conclusions are robust.

Appendices

Appendix A: A First-Order Approximation

Note that $|\eta _i|<M$, for $i=1,2,3,4$, which yields $|\bar{P}(x)|\le M<1$ for any x in [0, 1]. Using Taylor’s theorem, we obtain

$$\begin{aligned} \frac{1}{1+\bar{P}(x)}=1-\bar{P}(x)+\bar{P}^2(x)+\frac{\bar{P}^3(x)}{(1+\xi )^3}, \end{aligned}$$

(84)

where $\xi$ is a random variable that depends on $\bar{P}(x)$ such that $\xi \in (0,\bar{P}(x))$ if $\bar{P}(x)>0$ or $\xi \in (\bar{P}(x),0)$ if $\bar{P}(x)<0$. This leads to

$$\begin{aligned} E\left[ \frac{1+P_C(x)}{1+\bar{P}(x)}\right]&=E\left[ \left( 1+P_C(x)\right) \left( 1-\bar{P}(x) +\bar{P}^2(x)+\frac{\bar{P}^3(x)}{(1+\xi )^3}\right) \right] \nonumber \\&=E\Big [\left( 1+P_C(x)\right) \left( 1-\bar{P}(x)+\bar{P}^2(x)\right) \Big ] +E\left[ \frac{(1+P_C(x))\bar{P}^3(x)}{(1+\xi )^3}\right] . \end{aligned}$$

(85)

If $\bar{P}(x)>0$, we have $1\le 1+\xi \le 1+\bar{P}(x)$, which leads to

$$\begin{aligned} \left| \frac{1}{1+\xi }\right| \le 1. \end{aligned}$$

(86)

If $\bar{P}(x)<0$, we have $0<1+\bar{P}(x)\le 1+\xi \le 1$, which leads to

$$\begin{aligned} \left| \frac{1}{1+\xi }\right| \le \left| \frac{1}{1+\bar{P}(x)}\right| \le \frac{1}{1-|\bar{P}(x)|}\le \frac{1}{1-M}, \end{aligned}$$

(87)

since $|\bar{P}(x)|\le M<1$. Combining these inequalities, we get

$$\begin{aligned} \frac{1}{(1+\xi )^3}\le \sup \left\{ 1,\frac{1}{(1-M)^3}\right\} =K, \end{aligned}$$

(88)

where K is a finite constant. Then, we have

$$\begin{aligned} \begin{aligned}&\left| E\left[ \frac{(1+P_C(x))\bar{P}^3(x)}{(1+\xi )^3}\right] \right| \le E\left[ \frac{|1+P_C(x)||\bar{P}^3(x)|}{(1+\xi )^3}\right] \\&\quad \quad \quad \le K E\left[ |1+P_C(x)||\bar{P}^3(x)|\right] . \end{aligned} \end{aligned}$$

(89)

On the other hand, by using condition (3), we have

$$\begin{aligned} E\left[ |1+P_C(x)||\bar{P}^3(x)|\right] =o(\delta ). \end{aligned}$$

(90)

We conclude that

$$\begin{aligned} E\left[ \frac{(1+P_C(x))\bar{P}^3(x)}{(1+\xi )^3}\right] =o(\delta ). \end{aligned}$$

(91)

Appendix B: Calculation of Summations

Using the elementary arithmetic identities

$$\begin{aligned} \sum _{i=1}^{n}i^4&=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}, \end{aligned}$$

(92a)

$$\begin{aligned} \sum _{i=1}^{n}i^3&=\frac{n^2(n+1)^2}{4}, \end{aligned}$$

(92b)

$$\begin{aligned} \sum _{i=1}^{n}i^2&=\frac{n(n+1)(2n+1)}{6},\end{aligned}$$

(92c)

$$\begin{aligned} \sum _{i=1}^{n}i&=\frac{n(n+1)}{2}, \end{aligned}$$

(92d)

we get

$$\begin{aligned}&\sum _{i=1}^{N-1}(N-i)m\Big (\frac{i}{N}\Big )\nonumber \\&\quad =A_3\sum _{i=1}^{N-1}(N-i)\frac{i^3}{N^3}+A_2\sum _{i=1}^{N-1}(N-i) \frac{i^2}{N^2}+A_1\sum _{i=1}^{N-1}(N-i)\frac{i}{N}\nonumber \\&\quad \quad +A_0\sum _{i=1}^{N-1}(N-i)\nonumber \\&\quad =A_3\frac{(N^2-1)(3N^2-2)}{60N^2}+A_2\frac{N^2-1}{12}+A_1\frac{N^2-1}{6}+A_0\frac{N(N-1)}{2}\nonumber \\&\quad =\frac{N^2-1}{2}\Bigg [\frac{3N^2-2}{30N^2}A_3+\frac{A_2+2A_1}{6}+\frac{N}{N+1}A_0\Bigg ] \end{aligned}$$

(93)

and

$$\begin{aligned}&\sum _{i=1}^{N-1}m\Big (\frac{i}{N}\Big )\nonumber \\&\quad =A_3\sum _{i=1}^{N-1}\frac{i^3}{N^3}+A_2\sum _{i=1}^{N-1}\frac{i^2}{N^2}+A_1\sum _{i=1}^{N-1}\frac{i}{N}+A_0\sum _{i=1}^{N-1}1\nonumber \\&\quad =A_3\frac{N^2(N-1)^2}{4N^3}+A_2\frac{N(N-1)(2N-1)}{6N^2}+A_1\frac{N(N-1)}{2N}+A_0(N-1)\nonumber \\&\quad =(N-1)\Bigg [\frac{N-1}{4N}A_3+\frac{2N-1}{6N}A_2+\frac{A_1}{2}+A_0\Bigg ]. \end{aligned}$$

(94)

Appendix C: Simulation Data

This appendix contains the simulation data in Cases 1 and 2 under a Moran model (Figs. 9, 10) and a Wright–Fisher model (Figs. 11, 12). For a population size going from 2 to 20 and parameter values $\delta =0.02$, $\mu _b=2$ and $\mu _c=1$, the fixation probabilities $F_C$ and $F_D$ are calculated from $10^6$ repeated runs for each value of the scaled variance $\sigma ^2$ in a uniform probability distribution.

Fig. 9

Simulation data under a Moran model in Case 1

Fig. 10

Simulation data under a Moran model in Case 2

Fig. 11

Simulation data under a Wright–Fisher model in Case 1

Fig. 12

Simulation data under a Wright–Fisher model in Case 2